Important Reminders
Last Lecture
Complete Description of Motion
Intro to angular motion
Lectures will be M 11-12, T&W 10-12, F 11-12.
Today
Check schedule on web for new times and rooms
for some recitations (all are still on Thursday).
Statics and dynamics of rotational motion
Important Concepts
Equations for angular motion are mostly identical to those
for linear motion with the names of the variables changed.
Switching of recitations will be permitted if you have
a conflict with another IAP activity.
Location where forces are applied is now important.
Contact your tutor about session scheduling
Rotational inertia or moment of inertia (rotational
equivalent of mass) depends on how the material is
distributed relative to the axis.
Students working with Stephane Essame reassigned.
For an extended object, all of the equations learned
last fall apply exactly without approximations to the
motion of the center of mass.
This is true whether or not an object is also rotating.
The two motions (linear position of the center of
mass and rotation around the center of mass) can
be considered separately, except for kinetic energy
where everything gets lumped into one equation.
Mastering Physics due this Wednesday at 10pm.
Pset due this Friday at 11am.
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Kinematics Variables
x
 Angle
 Velocity
v
 Angular velocity ω
 Acceleration a
 Angular acceleration α
 Force F
 Torque τ
 Mass M
 Moment of Inertia I
 Momentum p
 Angular Momentum L
d"
!=
dt
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Vector Nature Angular Motion
θ
 Position
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Torque
The vector for any angular quantity (θ, ω, α, τ, J)
points along the axis with the direction given by a
right-hand-rule.
Fingers curl in direction of θ, ω, α, τ, J, thumb points in the
direction of the vector
For most problems, all variables can be considered
either clockwise (CW) or counter-clockwise (CCW).
How do you make something rotate? Very intuitive!
Larger force clearly gives more “twist”.
Force needs to be in the right direction (perpendicular to a
line to the axis is ideal).
The “twist” is bigger if the force is applied farther away
from the axis (bigger lever arm).
!
!
!
In math-speak: ! = r " F
d! d 2"
#=
= 2
dt
dt
! = r F sin(# )
F
φ
Axis
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r
Torque is out
of the page
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More Ways to Think of Torque
Conditions for Equilibrium
Magnitude of the force times the component of the
distance perpendicular to the force (aka lever arm).
Magnitude of the radial distance times the
component of the force perpendicular to the radius.
Direction from Right-Hand-Rule for cross-products
and can also be thought of as clockwise (CW) or
counter-clockwise (CCW).
For torque, gravity acts at the center of mass.
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Equations for Dynamics
!
!
Same as before: !F = Ma
!
Same as before: !F = 0
It’s totally irrelevant where the forces are applied to an
object, only their direction and magnitude matters.
This gives one independent equation per dimension.
This is true for any axis. However, if all of the forces are
in the same plane (the only type of problem we will
consider in this class), you only get one additional
independent equation by considering rotation.
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Inertia in Rotation
Depends on how the mass is distributed. Mass
farther from the axis is harder to rotate.
 I = !mi ri = " r dm
The same object could have a different moment of
inertia depending on the choice of axis.
!
In the equation: !" = I# all three quantities need to
be calculated using the same axis (either a fixed
axle or the center of mass).
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In addition, this equation holds for an axis through the
center of mass, even if the object moves or accelerates.
As for statics, if all of the forces are in the same plane,
you only get one additional independent equation by
considering rotation.
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Very simple way to find moment of inertia for a large
number of strange axis locations.
Some simple cases are given in the textbook on
page 342, you should be able to derive those below
except for the sphere. Will be on formula sheet.
Hoop (all mass at same radius) I=MR2
Solid cylinder or disk I=(1/2)MR2
Rod around end I=(1/3)ML2
Rod around center I=(1/12)ML2
Sphere I=(2/5)MR2
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!
Parallel Axis Theorum
Most easily derived by considering Kinetic Energy
(to be discussed next week).
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!
Additional condition: !" = I#
Moment of Inertia
Depends linearly on the total mass
!
This gives one independent equation per dimension.
This is true for any fixed axis (for example, a pulley).
!
Additional condition: !" = 0
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Only the direction and magnitude of the forces matter.
c.m.
d
Axis 1
I1 = Ic.m. + Md2 where M is the total mass.
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